Learn How to Solve SubFactorials (Left Factorials) | Quick & Simple Explanation

[prbmax]

Personally, I appreciate that you explain every symbol and slow walk every example.

[tapologomabotho]

My first time to see sub-factorial. Nice!

[karanbirsoin]

First time in forty two years after my electrical engineering course became aware of Left Factorial! Thanks to your video…

[justabunga]

The formula for subfactorial is also !n=nint(n!/e)=floor(n!/e+1/2) (also denoted as !n=⌊n!/e⌉= ⌊n!/e+1/2⌋). nint(x) stands for the nearest integer function, and floor stands for the greatest integer function (also denoted as int(x)). For example, nint(1.5)=2 and nint(2.5)=2. For the greatest integer function, here is an example: floor(1.9)=1, floor(1)=1, and floor(-1.9)=-2. In this video here, if you plug in n=3, then nint(3!/e)=2 (can also be floor(3!/e+1/2)=2). !5 will also be 44. e is a mathematical constant, which is about 2.718... I think here that this formula in this video is much easier to calculate (i.e. if calculators weren't allowed).

[marcusdecarvalho]

«One can show that !n equals the nearest integer to n!/e, where n! denotes the factorial of n and e is Euler's number.»

[patcon]

I usually appreciate your videos and find your method helpful.
We always strive to improve, and honest criticism is critical for growth.
Please accept the following comments in that spirit.
Full disclosure: My knowledge of subfactorials comes from a quick read of the Wikipedia article on the topic. My comments about teaching style are based on 44+ years of teaching math and science.

teaching perspective
You never gave the "why" we should care about this. As you can see from multiple comments most people have never come across this idea of derangements, let alone the subfactorial name/notation. While math can be beautiful for its own sake, when we ignore its connection to the world around us we lose quite a lot. That's sad because there's a concrete example to which people could relate. "In how many ways can a teacher hand back student tests for peer grading so that no student gets their own test?" It's also known as the "hat check" problem where hats are given back randomly and nobody gets the correct hat, or a misdirected mail situation where nobody gets the correct letter.

mathematics perspective
You quite nicely showed through brute force why !3 is 2 based on the definition of derangement.
This is actually good and necessary since you never proved or justified any formula, just boldly stated three possible formulae.
You then showed that one of these formulae gave the right answer for n=3.
You then computed this formula for n=5 and asserted without justification that it was the correct answer. I concede that showing all 44 derangements for ABCDE would be tedious.
Perhaps it would have been better to do this first for n=4, where !4=9 and this is easy enough to demonstrate by listing.
Then move on to n=5, using the formula to compute !5=44 (as you did), then starting the listing of derangements and challenging the viewers to finish.

[spdas]

Wow! What a beautiful, step by step, explanation of great thoughts with subtle details ! It helped me work out a big combinatorial math problem ! Great work, sir . Hats off from India . Kindly provide the generalized proof *** .

[FizzoMain]

Thanks, Im from Malaysia and this is very helpful! Im doing SASMO and the competition is in 6 days!

[Fabio_oares]

If I understand correctly, we can simplify the formula a little bit. As K=0 and K=1 will always be 1 - 1, we may adjust the formula with K beginning equal 2, instead of 0

[dilip]

It is dearrangement !n=(n!/e)
1st one 3!/e = 2.21 round off to 2
2nd one is 5!/e= 44.14 round of to 44

[krishnannarayanan]

LEARNING FROM THIS IS (-1) TO THE POWER 1 IS ONE AS WRITTEN IN THE SIDE FOR FINDING !3. KEEP IT UP!!!!

[victorchoripapa]

I've never heard about the left factorial. Very very interesting video! Thanks professor👌🏽 Greetings💪🏽🇵🇾

[jeausi]

This looked so hard when i first saw pics and explanations abt subfactorials but this vid helped me out a lotttt!

[tomcruise]

Commendable work, you are doing on YouTube ❤️

[lkern]

Is there a double subfactorial? Thank you sir.

[raghunandansingh]

Dear i may call Sir. I heard of left factorial. Today i learned how it is solved. Thanks sir.

[higoramorim]

I never saw this operation before, marvelous applications

[MathZoneKH]

Very nice sir! I got something new today from you ❤️greeting from Cambodia 🇰🇭

[jupudivenkataramamohan]

Thank you sir. I got another new concept.

[dhrubajyotisarma]

I first see subfactorial by your video I learned new things about math